Lab 6 - Rotational Equilibrium
Introduction
Have you ever tried to pull a stubborn nail out of a board or develop your forearm muscles by lifting weights? Both these activities involve using a "lever-type" action to produce a turning effect or torque through the application of a force. The same torque can be produced by applying a small force at a larger distance (with more leverage) or by applying a larger force closer to the point about which the object has to rotate. These two examples are shown in Fig. 1. In the case of the hammer pulling the nail, a small force applied at the end of the handle translates into a larger force being exerted on the nail at a smaller distance from the point where the nail is fixed to the board. In the second example the weight on the palm of the hand is at a greater distance from the elbow. This requires the muscles to apply a larger force at a smaller distance, usually less than 5 cm from the elbow. These are both examples of lever action—force applied at a distance from a fulcrum or pivot point or axis of rotation. A force applied as described in the above examples results in a torque on a body. Torque usually produces a rotation of a body.Figure 1: Two examples of torque
Discussion of Principles
Torque is a measure of the turning effect of an applied force on an object, and is the rotational analogue to force. In translational motion, a net force causes an object to accelerate, while in rotational motion, a net torque causes an object to increase or decrease its rate of rotation. Torqueτ
is the product of the applied force and the perpendicular distance from the pivot point to the line of action of the force and is measured in units of N·m.
( 1 )
τ = Fr ⊥
r⊥
is sometimes called the "lever arm."
Note: Torque has the same units as work, i.e., force times distance. Torque and work, however, are entirely different physical concepts; the fact that they have the same units is a coincidence.
Calculation of torque
Consider the irregularly shaped two-dimensional object shown in Fig. 2a.Figure 2: Illustration of lever-arm concept
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1Sketch a line through the force. This dashed line in Fig. 2a represents the line of action of the force.
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2Draw a perpendicular line from the axis of rotation O to the line of action of the force. This line, marked d in Fig. 2, represents the lever arm r defined in Eq. (1)τ = Fr ⊥.
Figure 3: Dependence of lever arm on point of application of force
( 2 )
d = r sin θ
θ
is the angle between r
and F.
Equation (1) can now be written as
( 3 )
τ = rF sin θ
Net torque
If two or more forces are applied to an object, each force produces a torque. The rotation of the wheel shown in Fig. 4 is caused by the sum of the two torques.Figure 4: A wheel experiencing two torques
F1
will produce a positive or counterclockwise torque, while F2
will produce a negative or clockwise torque. For rotation about the center the magnitude of the net torque will be the algebraic sum of the two torques:
( 4 )
τtotal = F1d1 − F2d2
r
and F.
The right-hand rule gives the direction of the torque. Based on this rule positive torques, such as F1d1,
are directed out of the page, while negative torques, such as F2d2,
are directed into the page.
Definitions of equilibrium
Torque causes rotational motion with angular (or rotational) accelerationα
.
( 5 )
τnet = Iα
α
is the angular acceleration. This equation is the angular equivalent of Newton's second law:
( 6 )
Fnet = ma
( 7 )
τ = 0 | |
F = 0 | |
OBJECTIVE
The objective of this experiment is to learn to measure torque due to a force and to adjust the magnitude of one or more forces and their lever arms to produce static equilibrium in a meter stick balanced on a knife edge; use the conditions for equilibrium to determine the mass of the meter stick and the mass of an unknown object.Equipment
- Meter stick
- Knife edge
- Known masses of varying values
- Unknown mass
- Balance
Procedure
There are three parts to this experiment. In the first part, you will balance three forces on a meter stick and show that the net torque is zero when the meter stick is in equilibrium. In the second part you will balance the weight of the meter stick against a known weight to determine the mass of the meter stick. Finally you will use the principle of rotational equilibrium to determine the mass of an unknown object. All lever arm distances are measured from the knife edge, which serves as the point of support. You will be using rubber bands to hang the weights on the meter stick. Assume that the masses of the rubber bands are negligible.Procedure A: Balancing Torques
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1Balance the meter stick on the knife edge. The point at which the stick balances is the center of gravity of the meter stick. Enter this value on the worksheet.
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2Select two 200-gram masses and one 100-gram mass.
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3Refer to Fig. 5 and Fig. 6. Place a hanger at the 20-cm mark, a distancex1cm to the left of the center of gravity and place massm1 = 200 gon it.
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Place another hanger at the 65-cm mark, a distance x2cm to the right of the center of gravity and place a massm2 = 200 gon it.
- Enter these values in Data Table 1.
Figure 5: Three balanced torques
Figure 6: Photo of experimental set-up
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4Calculate the torques due tom1 and m2and enter these values in Data Table 1. Be sure to include the sign of the torques.
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5Using the appropriate sign for each torque we can write the condition for rotational equilibrium as
( 8 )
τ | |
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6Use Eq. (8)and the values of the torques due to
= m1gx1 − m2gx2 − m3gx3 = 0τ m1 and m2to predict the torque due tom3(including its sign) and enter this value in Data Table 1. Be sure to include the sign of the torques. -
7Use the predicted value of the torque due tom3to predict the position ofx3at which the third massm3must be placed to balance the meter stick. Enter this value on the worksheet.
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8Experimentally determine the positionx3 of m3and enter this value on the worksheet.
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9Compare the two values for the positionx3by finding the percent difference between the predicted and experimental values ofx3.See Appendix B.
Checkpoint 1:
Ask your TA to check your set-up and calculations.
Ask your TA to check your set-up and calculations.
Procedure B: Finding the Mass of a Meter Stick
For this part of the experiment you will use a 200-gram mass, the meter stick and the knife edge.-
10Move the knife edge to the 25-cm mark. You will notice that the meter stick is no longer in equilibrium. The unbalanced force is the weight of the meter stick acting at its center of gravity.
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11Experimentally find the position,x1of the 200-gram mass, needed to balance the meter stick. Enter the value ofx1on the worksheet.
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12In the space provided on the worksheet, sketch and carefully label a diagram of the meter stick and the 200-gram mass. Show all the torque-producing forces. Remember that the weight of the meter stick acts at its center of gravity. Indicate on your diagram the directions (clockwise or counterclockwise) of each torque.
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13Calculate the torque due to the 200-gram mass and enter this value in Data Table 2.
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14Use the value of the torque due to the 200-gram mass and the conditions for rotational equilibrium to determine the torque due to the massm2of the meter stick. Enter this value in Data Table 2.
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15Using the value of the torque determined in step 14, calculate the value of the mass of the meter stick m2. Enter this value on the worksheet.
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16Use the balance to measure the mass of the meter stick.
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17Compare the measured and calculated values of the mass of the meter stick by computing the percent difference.
Checkpoint 2:
Ask your TA to check your diagram, set-up and calculations.
Ask your TA to check your diagram, set-up and calculations.
Procedure C: Determining an Unknown Mass
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18Position the center of gravity of the meter stick over the support.
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19Place a 50-gram massm1at the 70-cm mark and a 200-gram massm2at the 20-cm mark.
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20You will tie the free end of the string to a shot bucket around the 1-cm mark and hang it over the pulley as shown in Fig. 7 and Fig. 8.
Figure 7: Set-up for determining an unknown mass
Figure 8: Photo of set-up for determining an unknown mass
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21In the space provided on the worksheet, sketch and carefully label a diagram of this set-up. Show all the torque-producing forces. Indicate on your diagram the directions (clockwise or counterclockwise) of each torque.
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22Calculate the torques due tom1 and m2,and enter these values in Data Table 3.
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23Use the values of the torques due to the two masses and the conditions for rotational equilibrium to determine the torque due tom3.Enter this value in Data Table 3.
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24Now add small masses to the bucket until the stick balances.
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25Determine the massm3of the shot and bucket using a balance.
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26Compute the percent difference between the experimental and predicted values for the mass of the shot plus bucket.
Checkpoint 3:
Ask your TA to check your set-up, diagram and calculations.
Ask your TA to check your set-up, diagram and calculations.